Séminaire d’analyse fonctionnelleÉcole Polytechnique

W. B. JOHNSONBanach spaces all of whose subspaces have the approximation propertySéminaire d’analyse fonctionnelle (Polytechnique) (1979-1980), exp. no 16, p. 1-11<http://www.numdam.org/item?id=SAF_1979-1980____A13_0>

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SEMINAIRE

D A N A L Y S E FONCTIONNELLE

1979-1980

BANACH SPACES ALL OF WHOSE SUBSPACES HAVE

THE APPROXIMATION PROPERTY

W. B. JOHNSON

(Ohio State University)

ÉCOLE POLYTECHNIQUE

CENTRE DE MATHÉMATIQUES91128 PALAISEAU CEDEX - FRANCE

Tél. : (1) 941.82.00 - Poste N°

Télex : ECOLEX 691596 F

Exposé No XVI 29 Février 1980

XVI.1

INTRODUCTION

Let X be a Banach space all of whose subspaces have the app roximation property.A consequence of a recent result of Szankowski [19] (cf. also Theorem l.g.6. in

[16]) is that X is of type p for all p 2 and of cotype q for all q &#x3E; 2.

In terms of inequalities, this means that for each there areconstants 0 A B m so that

-

q P

holds for all finite seauences (x.) of vectors in X. If A q-1

and B p

.

1 q Premain bounded as p 2 , q ; 2 ; i. e. , if (*) holds for p = 2 = q , then Xis isomorphic to a Hilbert space by a result of Kwapien [11]. Szankowski’sresult thus gave support to the conjecture that X must be isomorphic to aHilbert space.

In this note we give a simple criterion which guarantees that every subspace ofevery quotient of a given space has the approximation property - even a muchstronger property called the uniform projection approximation property (u.p.a.p.,in short). The interest in such a criterion stems from the fact that there areBanach spaces which satisfy it and are not isomorphic to Hilbert spaces. One

oo kn .

such example; namely, ( E J 2 for appropriately chosen k n co and p n -*2n=1 . n

has already appeared as Example 1.g.7 in [15] with essentially our original proofthat every subspace has the u. p. a. p.. It takes some additional work to check thatevery subspace of every quotient has the u.p.a.p., so we briefly comment on thiskind of space at the beginning of section 2. However, the emphasis in section 2is on a more interesting class of examples which do not contain isomorphic copies

of 1,2 and which show that the characterization of Hilbert space given by le s haveLindenstrauss and Tzafriri [13] cannot be improved. Yoreover, these examples have

the property that every subspace of every quotient has a Schauder basis.

We use standard Banach space theory notation, as may be found in [15] and [16].Let us just mention that BE is the closed unit ball of the Banach space E.

This eminar will appear in the same form in the proceedings of a conférence

hèld in Bonn in Octoberj 1979.

XVI.2

1. CRITERION FOR THE U.P.A.P.

A Banach space X is said to have the uniform appoximation property -u. a. p. -(respectively, uniform projection approximation property - u.p. a.p. providedthere is a uniformity function f: 5~ + :IR + and a constant K 00 so that for

every finite dimensional subspace E of X , there is an operator (respectively,projection) T on X so that BBT BB K , Te = e for all e E E , anddim TX f(dim E).

The u.a.p., introduced in [181, is a stronger and quantitative version of thebounded approximation property. There are some interesting classes of spaceswhich possess even the u.p. a. p. , such as L p spaces [18] and reflexive Or liez

spaces [1~]. Heinrich [61, [7] used ultraproduct techniques to check that theu.a.p. is a self-dual property. From this result and the technique in [9] itfollows easily (see Proposition 1.1) that the u.p.a.p. is also self-dual.

The main result of [13] is that if X is a Banach which has the u.p.a.p. with

uniformity function f(n) = n (or, equivalently, f(n) = n + m for some constant

m), then X is isomorphic to a Hilbert space. In section 2 we whow that for any

fonction g: for which 0 g(n) - n .. c:o, there is a Banach space X

such that every subspace of every quotient of X and X* has the u.p.a.p. with

uniformity function g , and yet £ is not isomorphic to a subspace of any

quotient of X.

Propositions 1.1. A Banach X has the u.p. a.p. if and only if X* has the

~

’

Proof : assume that X has the u.p.a.p. Then by [7 J, X* has the u, a.p. , so we

can suppose that X ( resFectively, X’~~ has the u.p.a.p. (respectively, u.a.p.)with the same uniformity function f and ,for a certain constant K m .

Now suppose E c X , G C X* , dim E = dim G = n. We use the technique of [9] to

construct a projection P on X so that P = I , )(P]Î depends only on K ,G Gand dim PX is a function only of n . To do this, choose an operator T on X*so that T- I , K , and dim TX f(n) . By the version of the principleG G -

of local reflexivity proved in we can assume that T = S* for some operatorS on X. Now choose a projection Q on X so that Qx = x for EUSX,

)jQ)1 K , end dim f(n + f(n) ). Using the identities Q2 = Q and QS = S ,one easily checks that P S + Q - SQ is a projection onto QX Since SP = S , also follows that

Conversely, if X* has the u.p.a.p., then so does X** by the first part of theproof, hence X does also by the original version [12] (or see Lemma 1. e. 6. in

[15]) of the principle of local reflexivity. 0

We come now to the definition which yields a crittrion for all subspaces of agiven space to have the u.p.a.p.

XVI.3

Definition 1.2. Given a Banach space X , positive integers :~ and m , a

constant K , we say that X satisfies C(nJm,K) provided that there is ann-codimer.sional subspace Y of X so that every subspace E of Y with

dim E m is the range of a projection P from X with

Proposition 1. 3. Suppose X satisfies C( n, m, K) and Z is a subspace of X.

If F is e subspace of 7 with dim F m - 5n , then there is a subspace G of

Z so that dim G and G is the range of a projection Rfrom Z with S. 4K + 3

-

Proof: We make use of the following well-known (see, e. g. , p. 112 in [16] ) fact:

Fact 1. + : Let Q: X - W be a quotient mapping and sum-Dose E is a subspace ofW with dim E = n. Then there is a subspace G of X with dim G 5 so

that BL, c 3 QBG ç E ° - - --

Let Y be an n-codimensional subspace of X which satisfies the conditions inthe definition of C(n,m,K). Let Q: Z + be the quotient mapping and

select a subspace G of Z with dim G 5n so that

By replacing Q with G + F , we can assume that F V G (but now we know onlythat dim G 5n + dim F m) . By the C(n,m,K) condition on X , there is a

projection P from Z (even from X) onto G n Y with K . Therestriction of Q to (I- P)G is one-to-one and onto, since ker and thus

is well-defined. Now

Next we mention a property which is closely related to C( ’) but is easier tocheck and work with.

Definition 1.~: Given a Banach spaC9 X , positive integers n and m , and aconstant K, we say that X satisfies H(n,m.K) provided that thei-e is anry-codimensional subspace Y so that every subsp ace E of Y with_dim E m is K-Euclidean; i e., is K-isomorchic ta a Hilbert s-oace.

----

XVI.4

If X satisfies H(n,m,K) and is of type 2 with constant a, , then by Maurey’ sextension theorem rl7l X also satisfies C(n,m,ÀK). On the other hand, if Xsatisfies C(n,m,K) then Theorem 6.7 of [41 yields that there is a constantM = M(K) so that X satisfies H( n, m, M) . Now the main examples in section 2are type 2 spaces, so that cC.) and H(’) are essentially equivalent propertiesfor them. However, in order to investigate quotients of our examples we need todualize the C( -) property. The substance of Proposition 1.8 is that the H( ·)property dualizes and implies the C( -) property even for spaces which are not oftype 2.

Recall that a subspace Y of X is said to be ~-norming. over a subspace Z ofX* provided

for each z e Z . This is equivalent to saying that the natural restrictionmapping R from Z to Y* (defined by (Rz)y = z(y) for z e Z, y e Y) isa X-isomorphism, or that the natural evaluation mapping T: Y-Z* (defined by(Ty)z = z(y) for y e Y , z e Z) satisfies

(The weak* closure can of course be eliminated if Y is reflexive, or if

dim Z oo and B-1 is replaced by any strictiy smaller number. )

Lemma 1. 6: Suppose Y is a subspace of X, Z is a subspace of X*, and Y is

X-norming over Z. If every n _dimensional subspace of Y is K-Euclidean.then every subspace E of Z with dim E n is 3ÀK-Euclidean and

complemented in X*.

Proof: Given E ~ Z with dim E n, define Q: X - E* by (Qx)e = e(x).Since Y ils. À-norming over Z and dim E ~ , the restriction of Q to Yis a quotient mapping up to constant À. Therefore. by Fact 1.4, there is a

subspace G of Y with so that

Thus E~ , whence also E , is 3 ~K-Euclidean.

The complementation follows from the next lemma:

Lemma 1.’~ : that G is a K-Euclidean subspace of X and G is

X-norming over a subspace E of X*. Then E is ÀK-compleaeenIed in X.

Proof: Define T:X~ ~ G* by Tx* = x* ~G . Then TE has an inverse , S , with

~, - Since G* is K-Euclidean. there is a projection P from G* onto

TE with ~~P~4 K. Hence Q = SPT is a projection from X* onto E . C

Proposition 1. 8: Suppose that X satisfies H’ n, m, K) and m &#x3E; Then X*

satisfies K) and K).

XVI.5

Proof: In view of Lemma 1. 6, we only need to observe that if Y is an

n-codimensional subspace of X , then y is 4-norming over some 5n-codimensi onalsubspace of X*. This is a consequence ’of Fact 1.4. To see this, let Q be

the quotient mapping from X onto X~Y , and choose a subspace E of X with

dim E 5n so that

We claim that Y is 4-norming over E~ . Indeed, if f e- E~ , "rB1 &#x3E; 1 , and

x r, BX with f(x) &#x3E; 1 , then we can choose e e3B sothat Qe=Qx . ° Thus

x - e e 4 By and f(x-e) = f(x) &#x3E; 1 . 0

2. THE EXAMPLES

Since property H(n,m,K) is clearly a hereditary property, Propositions 1.8 and1.3 imply that if m(n) is sufficiently large relative to n and there is a

constant K so that X satisfies H(n,m(n),K) for infinitely many n , then

every subspace of every quotient of X and X* has the u.p.a.p. The space

has this property if p ~ 2 and kn - - are chosen appropriately. The only

restriction is that, having chosen pi , ki for 1 i n , the pi 1 s for i &#x3E; ni i - - 1

must be chosen sufficiently close to 2 so that every subspace E of

is 2-Euclidean. If, having chosen p n+1 # 2 , one chooses k n+1 so that

then the resulting space

is not isomorphic to j’2. - (See p.112 in for one way cf carrying out this2

construction.)

XVI.6

By the results of [10] , every subspace of every quotient of such a space X is

isomorphic to a space of the form ( E En)2 with diin for every n . In

particular, every subspace of every quotient of such an X has an unconditionaldecomposition into finite dimensional subspaces. However not every subspace ofa space of the form

can have an unconditional basis unless X is isomorphic to 1 . k

Indeed, in2

k

[3] (or see [2 ] ) it is shown that there is a subspace E withn p n

dim E n = [k n ~2] so that

where c &#x3E; 0 is an absolute constant and glE is the Gordon-Lewis constant ofE ( see [2] ). From [5 J it follows that ( EE ) cannot have an unconditional basis

n2if

that is, if X is not isomorrhic to 12.using Kwapien’s characterization of Hilbert space [11J and the same reasoning

as above, one obtains the folloding:

Proposition 2.1: Suppose that is a sequence of Banach suaces. none of

which is isomorphic to a Hilbert suace. Xn has type pn and coty-pe qn with

constant K ( K independent of n) , and qn - p .. 0 as n - - . Then there are- n n -

finite dimensional subspaces E of X so that every subspace of every quotient

of (ZE n)2 has the u.p. a.p. , but is not isomorphic to 12. .

- n2 -

n 2 - ’

’2

We turn now to a more interesting class of examples.

Exemple 2.2: There is a constant ~i ~ so that if g(n) is a uositive integervalued function for which gf n) t CD, then there is X = X(g)

CI:)

2. 1 X has a unconditional basis

2. ~. X is of tvpe 2 and of q for all q &#x3E; 2.

(2.5) 1 2 is not isomorphic to a subspace of a quotient of X .

(2.6) Let Y be any subs-oace of a auotient ()f X or X*. If F is a subsjace

of Y with dim F x rn. there are F ç G I àim E &#x3E; m - 9

XVI.7

dim G m + g~m) , so that E is M-Euclidean and both J and G are

M-complemented in Z. - - -

(2.7 ) Every subspace of every quotient of X has a Schauder basis.

X = X(g) is the 2-convexification (see p. 53 of [16]) of the spacespanned by a suitable subsequence of the basis construc ted in [8]. The con-struction of this space is given in Example j. 3 of [~]. X can be described asthe completion of the space xo of finitely non-zero séquences of scalars under

a certain norm f ~ ·~~ . - The unit vectors form a monotonely unconditional basis

basis for X. For a certain increasing sequence of positive integersn n=

which depends on g , the norm H.BB satisfies property (2.8) below; in fact,

~l - ~~ is the unique norm on X 0which satisfies ~ 2. 8) . (For x e X o andA ç let Ax denote the sequence which agrees with x for coordinates in

A and is 0 el.sewhere. ~ - k

where the sup is over all n and all pairwise disjoint sequences

subsets of w+ for which

It follows from [8] (see section 1+ of [1] ) that L2 does not embed into X. X

is 2-convex with constant 1 since it is the 2-convexification of the space con-k

structed in [8]. It is also clear that if are disjointly supportedi i=l -

vectors in , then ,

Thus if k r.

t a) sufficiently quickly, (X, BB. BB) will satisfy a lower ,~ q -estimatefor every q &#x3E; 2 ; · that is, BBr:x.1B &#x3E; for all disjoint vectors (x.)for every q &#x3E; 2 ; that is, E 1 il for all disjoint vectors

1 ilin X and some constant c = c(q) &#x3E; 0. Thus (see section 1. f of [16] ) X has

type 2 and cotype q for all q &#x3E; 2 .

To see that (2.j ) is true it is enough to observe that 1, does not embed into

X*. Indeed, if "2 embeds into a quotient Y of X , thenf .2 is complemer:ted’ 2 in Y because Y is of type 2(cf . [17J); hence 2 is isomorrhic to a quotientof X , whence to a subspace of X’~ . But X is reflexive (since (2.9) precludesits containing a copy of co or il so that James’ theorem applies; cf. Theorem

1. c.12 in [15 ) ) so X* has a 2-concave basis. Now if £~ embeds into X* , then

some block basis of the 2-cJncavJe basis for X* is equivalent ta the unit vectorbasis for 92 and thus ( see, e. g. , the argument for Theorem 3.1 in [18l ) spans a

XVI.8

complemented subspace. This would imply that P 2 embeds into X-~- = X , which

is false.

We now turn to the proof of (2.6).

From the argument for Theorem 2.1 of [181 (see Proposition 3 of [141 for a sketchof the proof in the generality we need) it follows that if H is an n-dimensionalsubspace of a space which has a donotonely unconditional basis, then H is

2-isomorphic to a subspace spanned by. 1(n) disjointly supported vectors, where

1(n) dépends onl y on n . (J!(n) is in fact of ord e r Therefore, if .

00H is a subspace of the subspace of X spanned by dim H = s and

,~( s ) kn 19 then by (2. 9) H is 4-Euclidean. This means that for arbitrary d nX satisfies H(n,d n ,h) if kn is sufficiently large. Thus by Proposition 1.8

and obvious duality arguments we have: ,

There is a constant K so that for any sequence dn t - there is a

sequence kn t = so that if Y is a subspace of a quotient of X or X~

(2.10) (where X = X(k ) satisfies (2.8)), then for each n = 1,2,..., Yn

satisfies H(n,d n K) and C(n,d n K) for the same n-codimensional subspace

Yn of Y.

Suppose now that for each n = 1, 2, ... , Y satisfies H(n,d ,K) and C(n,d ,K)for the same n-codimensional subspace Yn of Y. Let F be any m-dimensional

subspace of Y and suppose that n satisfies .

If also

then E = F f1 Yn fulfills the conditions in ~2. ~~ j for M = K .

On the other hand, if

then by Proposition 1.3 there is a subspace G of Y so that F ~ G ,

dim G m + and G is 4K + 3 - complemented in Y . This G fulfills theconditions in (2.6) for M = 4K + 3 as long as

This complètes the rroof of (2. 6), because for any sequence g(m) t ~ we can select

dn t - so that for every m there is an n for which (2.11)-(2.1~) are satisfied.

To prove (2.7), we use the technique of [9]. By the proof of (2.6), we can assumethat the space X has the property that there is a constant M so that if Z is

any subspace of a quotient of X or of X* , then for all n = 1,2, ... we have :

C2.15 ~ Z satisfies H(n,3~BM)

(2.16) If E is a subspace of Z then there is a projection P on Z so

B . ~ that Pe = e for e ~ E , M , and dim PZ 1.01 dim E.

XVI.9

Let Y be a subspace of a quotient of X , and for n = 1.2, ... let Yn be a

codimensional n subspace of Y as in the definition of H(n,3n+ 1 ,M). · We

use (2.1~) and (2.16) ta construct a sequence 1 of projections on Y to’

n n_ 1

satisfy the following conditions for all 1 n,m ~ :

/

Conditions (2.17)-2.20) are standard conditions which guarantee that

En = (p - n (where Po = 0) forms a finite dimensional decomposition for

Y with dim E 3n . Now (2.21) implies that E is M-Euclidean by (2.15).n - n+

sTherefore we can select a sequence in En+1 (where so = 0 andi 1= s + n+1 a

n

sn = dim E1 + E2 +...+En for n &#x3E; 1) which is M-equivalent to an orthonormal

sequence in a Hilbert space. The,sequence is thus a basis for Y . 8

co m

It remains to construct the sequence (p) n ll= 18 Let l be dense in Y with

x 1 = 0. Choose f e Y 1 with lif = 1, and y ~ Y with Ilyll = Ilf Il =f(y) , and

set Pl = f y. Y has codimension one in Y , so span(f) = Y1 or ker f = Yand hence (I - Thus (2.17) and (2.1g)-(2.c^1) are satisfied for

n = i = m and *

Having constructed P1,...,Pk to satisfy (2.17) and (2.19)-(2.21) for all

1 n, m k and P x. = x. we construct Pk+1 as follows :n 1 1 201320132013’ k+l

By (2.16) and the reflexivity of Y , there is a projection P on Y so that

p*g = g for g e Y1 M , and dim P Y Again byk+1 k y - -

(2.16), y there is a projection Q on Y so that Qy = y for y e

M , and

Using the identifies and

we have that P is a projection onto cY , and thus

for and for 1 x . k+1. Since- 1.-

, we have that ThereforeIl Il fj

for m and (are satisfied for , and 1

XVI.10

It is clear that the canstrllcted sequence In ° (.21). Cln n...

Remarks ; 1. We do not know whethe r the X of can be constructedso that each o.f its su?&#x3E;spaces has an unconditional- basis.

2. We do nct know wliether there is a space wit,h a synm1etric basis(other than 1, ) or a non-reflexi ve spncc such thR t cvery subspace has the

approximation property. If the space Z is not isomorphic to 1,, but satisfiesri

C(n, dn, K) for infinitely mal1Y n and sorne d n then certainly Z cannot

have a symmetric or even subsymmetric basis, and Z must be super-reflexive.

XVI.11

REFERENCES

1. T. Figiel and W. E. Johnson, A uniformly convex Banach space which contains no

lp, Compositio Math. 29 (1974), 179-190.

2. T. Figiel and W.B. Johnson, Large subspaces of ln~ and estimates of the

Gordon-Lewis constant, Israel J. Math. (submitted)

3. T. Figiel, S. Kwapien, and A. Pelczynski, Sharp estimates for the constantsof local unconditional structure of Minkowski spaces, Bull. Acad. Polon. Sci.

25(1977), 1221-1226.

4. T. Figiel, J. Lindenstrauss, and V. Milman, The dimension of almost sphericalsections of convex bodies, Acta Math. 139(1977), 53-94.

5. Y. Gordon and D.R. Lewis, Absolutely summing operators and local unconditionalstructures, Acta Math. 133(1974), 27-48.

6. S. Heinrich, Finite representability and super ideals of operators, Disser-tationes Math. ( to appear).

7. S. Heinrich, Ultraproducts in Banach space theory, Arkiv für Reine und Angew.Math. ( to appear).

8. W. B. Johnson, A reflexive Banach space which is not sufficiently Euclidean,Studia Math. 55(1976), 201-205.

9. W.B. Johnson, H.P. Rosenthal, and M. Zippin, On bases, finite dimensional

decompositions, and weaker structures in Banach spaces, Israel J. Math. 9(1971), 488-506.

10. W. B. Johnson and M. Zippin, On subspaces of quotients of (03A3G n)lp and

(03A3Gn)co , Israel J. Math. 13 (1972), 311-316.

11. S. Kwapien, Isomorphic characterization of inner product spaces by orthogonalseries with vector valued coefficients, Studia Math. 44(1972), 583-595.

12. J. Lindenstrauss and H.P. Rosenthal, the Lp

spaces. Israel J. Math. 7 (1969),

325-349.

13. J. Lindenstrauss and L. Tzafiri, On the complemented subspaces problem,Israel J. Math. 9 (1971), 263-269.

14. J. ’Lindenstrauss and L. Tzafriri, The uniform approximation property in Orliczspaces, Israel J. Math. 23(1976), 142-155.

15. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I, Sequence spaces,Ergebnisse 92; Springer-Verlag (1977).

16. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II, Function spaces,Ergebnisse 97; Springer-Verlag (1979).

17. B. Maurey, Un théorème de prolongement. C.R. Acad. Sci. Paris, Sér A-B 279(1974), 329-332.

18. A. Pelczynski and H. P. Rosenthal, Localization techniques in Lp spaces,Studia Math. 52(1975), 263-289.

19. A. Szankowski, Subspaces without approximation property, Israel J. Math. 30

(1978), 123-129.